## Conservation of Energy First Principle of Thermodynamics

30 The first principle of thermodynamics relates the work done on a (closed) system and the heat transfer into the system to the change in energy of the system. We shall assume that the only energy transfers to the system are by mechanical work done on the system by surface traction and body forces, by heat transfer through the boundary. 6.4.1 Spatial Gradient of the Velocity 31 We define L as the spatial gradient of the velocity and in turn this gradient can be decomposed into a symmetric rate...

## Traction on an Arbitrary Plane Cauchys Stress Tensor

7 Let us now consider the problem of determining the traction acting on the surface of an oblique plane (characterized by its normal n) in terms of the known tractions normal to the three principal axis, 11, t2 and t3. This will be done through the so-called Cauchy's tetrahedron shown in Fig. 2.3. 8 The components of the unit vector n are the direction cosines of its direction n1 cos( Z AON) n2 cos(Z BON) n3 cos(Z CON) (2.3) The altitude ON, of length h is a leg of the three right triangles...

## Force Traction and Stress Vectors

1 There are two kinds of forces in continuum mechanics body forces act on the elements of volume or mass inside the body, e.g. gravity, electromagnetic fields. dF pbdVol. surface forces are contact forces acting on the free body at its bounding surface. Those will be defined in terms of force per unit area. 2 The surface force per unit area acting on an element dS is called traction or more accurately stress vector. I tdS if txdS j tydS k tzdS 2.1 Most authors limit the term traction to an...

## Load Shear Moment Relations

38 Let us derive the basic relations between load, shear and moment. Considering an infinitesimal length dx of a beam subjected to a positive load5 w x , Fig. 12.6. The infinitesimal section must also be in equilibrium. 39 There are no axial forces, thus we only have two equations of equilibrium to satisfy XFy 0 and XMz 0. 40 Since dx is infinitesimally small, the small variation in load along it can be neglected, therefore we assume w x to be constant along dx. 41 To denote that a small change...

## Principle of Virtual Work and Complementary Virtual Work

33 The principles of Virtual Work and Complementary Virtual Work relate force systems which satisfy the requirements of equilibrium,, and deformation systems which satisfy the requirement of compatibility. 1. In any application the force system could either be the actual set of external loads dp or some virtual force system which happens to satisfy the condition of equilibrium p. This set of external forces will induce internal actual forces da or internal hypothetical forces Sa compatible with...

## Geometric Instability

33 The stability of a structure is determined not only by the number of reactions but also by their arrangement. 34 Geometric instability will occur if 1. All reactions are parallel and a non-parallel load is applied to the structure. 2. All reactions are concurrent, Fig. 12.4. Figure 12.4 Geometric Instability Caused by Concurrent Reactions Figure 12.4 Geometric Instability Caused by Concurrent Reactions 3. The number of reactions is smaller than the number of equations of equilibrium, that is...

## Example Mohrs Circle in Plane Stress

An element in plane stress is subjected to stresses axx 15, ayy 5 and Txy 4. Using the Mohr's circle determine a the stresses acting on an element rotated through an angle 0 40 counterclockwise b the principal stresses and c the maximum shear stresses. Show all results on sketches of properly oriented elements. Solution 1. The center of the circle is located at 2 axx ayy 1 15 5 10. 2.63 2. The radius and the angle 2p are given by R y1 15 5 2 42 6.403 2.64-a 2 4 tan2 w 0.8 38.66 f3 19.33 2.64-b...

## Boundary Conditions

4 In describing the boundary conditions B.C. , we must note that 1. Either we know the displacement but not the traction, or we know the traction and not the corresponding displacement. We can never know both a priori. 2. Not all boundary conditions specifications are acceptable. For example we can not apply tractions to the entire surface of the body. Unless those tractions are specially prescribed, they may not necessarily satisfy equilibrium. 5 Properly specified boundary conditions result...

## Isotropic Material

40 An isotropic material is symmetric with respect to every plane and every axis, that is the elastic properties are identical in all directions. 41 To mathematically characterize an isotropic material, we require coordinate transformation with rotation about x2 and xi axes in addition to all previous coordinate transformations. This process will enforce symmetry about all planes and all axes. 42 The rotation about the x2 axis is obtained through cos 6 0 sin 6 0 1 0 sin 6 0 cos 6 we follow a...

## V

Finally, the displacement components can be obtained by integrating the above equations 10.2.2.3 Example Thick-Walled Cylinder 4i If we consider a circular cylinder with internal and external radii a and b respectively, subjected to internal and external pressures p and po respectively, Fig. 10.2, then the boundary conditions for the plane strain problem are Ttt Pi at r Oi Ttt -po at r b 42 These Boundary conditions can be easily shown to be satisfied by the following stress field These...

## Rayleigh Ritz Method

55 Continuous systems have infinite number of degrees of freedom, those are the displacements at every point within the structure. Their behavior can be described by the Euler Equation, or the partial differential equation of equilibrium. However, only the simplest problems have an exact solution which satisfies equilibrium, and the boundary conditions . 56 An approximate method of solution is the Rayleigh-Ritz method which is based on the principle of virtual displacements. In this method we...

## Arch Elasticity

51 Fig. 2.10 illustrates the stresses acting on a differential element of a shell structure. The resulting forces in turn are shown in Fig. 2.11 and for simplification those acting per unit length of the middle surface are shown in Fig. 2.12. The net resultant forces are given by Figure 2.9 Mohr Circle for Stress in 3D Figure 2.10 Differential Shell Element, Stresses Figure 2.10 Differential Shell Element, Stresses Figure 2.11 Differential Shell Element, Forces Figure 2.11 Differential Shell...

## Spherical and Deviatoric Stress Tensors

36 If we let o denote the mean normal stress p 0 -p 3 011 022 033 3on tr a then the stress tensor can be written as the sum of two tensors Hydrostatic stress in which each normal stress is equal to p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. Deviatoric Stress which causes the change in shape.

## Navier Cauchy Equations

11 One such approach is to substitute the displacement-strain relation into Hooke's law resulting in stresses in terms of the gradient of the displacement , and the resulting equation into the equation of motion to obtain three second-order partial differential equations for the three displacement components known as Navier's Equation Figure 9.3 Fundamental Equations in Solid Mechanics

## Shear Moment and Deflection Diagrams for BEAMS

Adapted from 1 Simple Beam uniform Load 2 Simple Beam Unsymmetric Triangular Load 3 Simple Beam Symmetric Triangular Load 3 Simple Beam Symmetric Triangular Load 4 Simple Beam Uniform Load Partially Distributed 4 Simple Beam Uniform Load Partially Distributed 5 Simple Beam Concentrated Load at Center at x 2 when x lt 2 whenx lt 2 at x L 6 Simple Beam Concentrated Load at Any Point

## Strain Decomposition

72 In this section we first seek to express the relative displacement vector as the sum of the linear Lagrangian or Eulerian strain tensor and the linear Lagrangian or Eulerian rotation tensor. This is restricted to small strains. 73 For finite strains, the former additive decomposition is no longer valid, instead we shall consider the strain tensor as a product of a rotation tensor and a stretch tensor. 4.3.1 jLinear Strain and Rotation Tensors 74 Strain components are quantitative measures of...

## Euler Equation

1 The fundamental problem of the calculus of variation1 is to find a function u x such that 2 We define u x to be a function of x in the interval a, b , and F to be a known function such as the energy density . 3 We define the domain of a functional as the collection of admissible functions belonging to a class of functions in function space rather than a region in coordinate space as is the case for a function . 4 We seek the function u x which extremizes n. 5 Letting u to be a family of...

## Principle of Virtual Work

35 Derivation of the principle of virtual work starts with the assumption of that forces are in equilibrium and satisfaction of the static boundary conditions. The Equation of equilibrium Eq. 6.26 which is rewritten as where b representing the body force. In matrix form, this can be rewritten as Note that this equation can be generalized to 3D. 37 The surface r of the solid can be decomposed into two parts r and T where tractions and displacements are respectively specified. t t on rt Natural...

## Principal Strains Strain Invariants Mohr Circle

So Determination of the principal strains E 3 lt E 2 lt E i , strain invariants and the Mohr circle for strain parallel the one for stresses Sect. 2.4 and will not be repeated here. where the symbols IE, IIE and IIIE denote the following scalar expressions in the strain components Ie En E22 E33 En tr E 4.164 IIe E11E22 E22E33 E33E11 E223 E31 E222 4.165 2 EjEij EiiEjj 2Eij Eij 2IE 4.166 IIIe detE eijk epqr EipEjqEkr 4.168 87 In terms of the principal strains, those invariants can be simplified...

## Statics Equilibrium

8 Any structural element, or part of it, must satisfy equilibrium. 1 So far we have restricted ourselves to a continuum, in this chapter we will consider a structural element. Summation of forces and moments, in a static system must be equal to zero2. In a 3D cartesian coordinate system there are a total of 6 independent equations of equilibrium SFx SFy SF 0 SMx SMy SM2 0 11 In a 2D cartesian coordinate system there are a total of 3 independent equations of equilibrium 12 All the externally...

## Principle of Complementary Virtual Work

40 Derivation of the principle of complementary virtual work starts from the assumption of a kinematicaly admissible displacements and satisfaction of the essential boundary conditions. 41 Whereas we have previously used the vector notation for the principle of virtual work, we will now use the tensor notation for this derivation. 42 The kinematic condition strain-displacement 43 The essential boundary conditions are expressed as 44 The principle of virtual complementary work or more...

## Coordinate Transformation jGeneral Tensors

9 Let us consider two bases bj x x3 and bj xi, x2x3 , Fig. 1.5. Each unit vector in one basis must be a linear combination of the vectors of the other basis bj- apbp and bk b bg 1.20 summed on p and q respectively where apP subscript new, superscript old and bk are the coefficients for the forward and backward changes respectively from b to b respectively. Explicitly Figure 1.5 Coordinate Transformation Figure 1.5 Coordinate Transformation 10 The transformation must have the determinant of its...

## Introduction

1 In Eq. we showed that around a circular hole in an infinite plate under uniform traction, we do have a stress concentration factor of 3. 2 Following a similar approach though with curvilinear coordinates , it can be shown that if we have an elliptical hole, Fig. , we would have We observe that for a b, we recover the stress concentration factor of 3 of a circular hole, and that for a degenerated ellipse, i.e a crack there is an infinite stress. Alternatively, the stress can be expressed in...

## Lagrangian Stresses Piola Kirchoff Stress Tensors

96 In Sect. 2.2 the discussion of stress applied to the deformed configuration dA using spatial coordiantes x , that is the one where equilibrium must hold. The deformed configuration being the natural one in which to characterize stress. Hence we had note the use of T instead of a . Hence the Cauchy stress tensor was really defined in the Eulerian space. 97 However, there are certain advantages in referring all quantities back to the undeformed configuration Lagrangian of the body because...

## Strain Tensor

8 Following the simplified and restrictive introduction to strain, we now turn our attention to a rigorous presentation of this important deformation tensor. 9 The approach we will take in this section is as follows 1. Define Material fixed, Xj and Spatial moving, Xj coordinate systems. 2. Introduce the notion of a position and of a displacement vector, U, u, with respect to either coordinate system . 3. Introduce Lagrangian and Eulerian descriptions. 4. Introduce the notion of a material...

## Cartesian Coordinates Plane Strain

16 If the deformation of a cylindrical body is such that there is no axial components of the displacement and that the other components do not depend on the axial coordinate, then the body is said to be in a state of plane strain. If e3 is the direction corresponding to the cylindrical axis, then we have Ui Ui xi,x2 , U2 U2 xi,X2 , U3 0 and the strain components corresponding to those displacements are and the non-zero stress components are Tii, Ti2, T22, T33 where 17 Considering a static...

## F Experimental Measurement of Strain

90 Typically, the transducer to measure strains in a material is the strain gage. The most common type of strain gage used today for stress analysis is the bonded resistance strain gage shown in Figure 4.9. Figure 4.9 Bonded Resistance Strain Gage 91 These gages use a grid of fine wire or a metal foil grid encapsulated in a thin resin backing. The gage is glued to the carefully prepared test specimen by a thin layer of epoxy. The epoxy acts as the carrier matrix to transfer the strain in the...

## Hydrostatic and Deviatoric Strain

85 The lagrangian and Eulerian linear strain tensors can each be split into spherical and deviator tensor as was the case for the stresses. Hence, if we define then the components of the strain deviator E' are given by We note that E' measures the change in shape of an element, while the spherical or hydrostatic strain iel represents the volume change. es disusing lt C Jo h0 we t,ai t tOlst MatrixForm Tfirst . 0, 1, 0 gt We note that this vector is in the same direction as t e Cauchy stress...